integral from 1 to 2 of e^{1/x}

Lösung

\int_{1}^{2} e^{\frac{1}{x}} d x

- e + Ei (1) + 2 e - Ei (\frac{1}{2})

Schritte zur Lösung

\int_{1}^{2} e^{\frac{1}{x}} d x

Wende U-Substitution an

= \int_{1}^{\frac{1}{2}} - \frac{e ^{u}}{u ^{2}} d u

\int_{a}^{b} f (x) d x = - \int_{b}^{a} f (x) d x, a < b

= - \int_{\frac{1}{2}}^{1} - \frac{e ^{u}}{u ^{2}} d u

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - (- \int_{\frac{1}{2}}^{1} \frac{e ^{u}}{u ^{2}} d u)

Wende die partielle Integration an

= - (- [- \frac{e ^{u}}{u} - \int - \frac{e ^{u}}{u} d u]_{\frac{1}{2}}^{1})

\int - \frac{e ^{u}}{u} d u = - Ei (u)

= - (- [- \frac{e ^{u}}{u} - (- Ei (u))]_{\frac{1}{2}}^{1})

Vereinfache

= [- \frac{e ^{u}}{u} + Ei (u)]_{\frac{1}{2}}^{1}

Berechne die Grenzen:

- e + Ei (1) + 2 e - Ei (\frac{1}{2})

= - e + Ei (1) + 2 e - Ei (\frac{1}{2})